January-February 1992By Gary Mar
Gary Mar is Assistant Professor of Philosophy at the State University of New York at Stony Brook
Does God Play Dice? : The Mathematics of Chaos. By Ian Stewart. Basil Blackwell. 348 pages. $10.95.
Chaos: Making a new Science. By James Gleick. Penguin. 352 pages. $9.95.
The Dreams of Reason: The Computer and the Rise of the Sciences of Complexity. By Heinz Pagels. Bantam. 352 pages. $9.95.
In the beginning God created order out of chaos. Across branches of science as diverse as physics and physiology, economics and ecology, a new genesis is taking place as scientists are discovering how chaos can arise out of order.
Ian Stewart, in Does God Play Dice? (now reissued in paperback), traces the history of chaos theory back to the work of the great French mathematician Poincare. The triumph of Newtonian mechanics had inspired Laplaces classic statement of determinism: Assume an intelligence which at a given moment knows all the forces that animate nature as well as the situations of all the bodies that compose it, and further that it is vast enough to perform a calculation based on these data . For it nothing would be uncertain, and the future, like the past, would be present before its eyes. Perhaps intrigued by such possibilities, King Oscar II of Sweden offered the equivalent of a Nobel Prize in 1887 for an answer to the question Is the universe stable? Two years later Poincare was awarded the prize for his celebrated work on the three body problem. Poincare showed that even a system comprised of only the sun, the earth, and the moon, but governed by Newtons law of gravity, could generate dynamic behavior of such incalculable complexity that prediction would be impossible. Just as Einsteins theory of relativity later eliminated the Newtonian idea of absolute space, Poincares discovery of chaos within the framework of classical Newtonian mechanics dispelled the Laplacian dream of deterministic predictability.
Poincares pioneering work revolutionized the study of nonlinear equations by replacing quantitative analytic methods with qualitative techniques from geometry and topology. Chaos theory is indeed sometimes called a nonlinear science. But as the mathematician Stanislaw Ulam wryly observed, this is like calling zoology the study of nonelephant animals. Nonlinear phenomena occur everywhere in nature, but textbooks on solving differential equations tended to ignore the nonlinear except in those cases in which linear approximations could be substituted for the nonlinear. The nonlinear was nearly synonymous with the unsolvable, and so physicists learned not to see chaos.
The story of how the M.I.T. meteorologist Edward Lorenz learned, almost by accident, to see chaos is retold in James Gleicks Chaos: Making a New Science, a bestseller now reissued in paperback, which chronicles the recent history of chaos theory with the excitement of a novel. Lorenz had constructed a simulation of the weather on his Royal McBee computer, which ran on vacuum tubes. One winter day in 1961, in order to save time, Lorenz, instead of running his computer from the beginning, simply typed the numbers straight from the printout midway through a run. To his surprise, Lorenz found that the new run diverged radically from the original even though the equations were completely deterministic. Lorenz checked his inputs and found he had made no mistakes. What had gone wrong? It turned out that the printout only listed the outputs to three significant digits whereas the computer carried the computations to six: A difference of only one one-thousandth was enough to doom predictability. Under the influence of Newton, physicists assumed that with approximate knowledge of a systems initial conditions and the laws of nature, one could calculate the approximate behavior of the system. That day Lorenz realized this assumption was false: The average person, seeing that we can predict tides pretty well a few months ahead would say, why cant we do the same thing with the atmosphere . But I realized that any physical system that behaved nonperiodically would be unpredictable. Lorenz later dubbed this extreme sensitivity to initial conditions the Butterfly Effect, to stand for the metaphorical idea that a butterfly stirring the air today in Brazil could cause a tornado next month in Texas.
Sensitivity to initial conditions was later taken to be the quintessential element of a mathematical definition of chaos set forth in a 1975 paper intriguingly titled Period Three Implies Chaos by Tien Li and James Yorke. Li and Yorke proved that any one-dimensional system which had a cycle of period three will also display cycles of every other degree as well as completely chaotic cycles. It turned out that the Russian mathematician Sarkovskii had proved the result before them, but Yorke had sent the message to the physicists. The discovery that chaos was ubiquitous, structured, and stable came as an electric shock to physicists like Freeman Dyson. Previously it had been thought that chaos arose because of an unwieldy number of parameters. Now physicists knew that chaos must arise in systems governed even by the simplest of nonlinear equations.
Not only can chaos be mathematically studied, it can be seen. Computers revealed that deep within chaos are hidden patterns known as fractals intriguing objects that exhibit infinitely complex self-affinity at increasing powers of magnification. Fractals are descendants of certain turn of the century pathological monstrosities such as the one-dimensional space filling the von Koch snowflake, which has a finite area but an infinitely complex border. Galileos claim that the Euclidian geometry was language of nature was ridiculed in Jonathan Swifts Gullivers Travels in 1726, but the view was mathematically refuted in 1982 in Benoit Mandelbrots The Fractal Geometry of Nature. As Freeman Dyson ironically observed, Nature has played a joke on mathematicians. The 19th-century mathematicians may have been lacking in imagination, but nature was not. The same pathological structures that the mathematicians invented to break loose from 19th-century naturalism turned out to be inherent in familiar objects all around us.
The fractal world was previously inaccessible not because fractals were too small or too far away, but because they are too complex for any human mind to visualize them. In The Dreams of Reason (also now reissued in paperback), Heinz Pagels insightfully explores the role of the computer as a research instrument as it cracks the complexity barrier across the scientific disciplines, and the reordering of knowledge this implies for the philosophy of science.
Ever since the rise of modern science three centuries ago, the instruments of investigation such as telescopes and microscopes have promoted a reductionist view of science. Physics was deemed the most fundamental science because it dealt with the smallest and most reduced entities. From the laws of physics one was supposed to deduce the laws of chemistry, then of life, and so on up the ladder to consciousness itself. Pagels does not claim that this view of nature is wrong, but only that the view has been powerfully shaped by available technology. (The view is not novel; the influence of the telescope on the imagery of Miltons Paradise Lost, for example, has long been noted by literary critics.) The computer is now forcing us to revise this view of nature because the computer produces knowledge differently from traditional analytic instruments like the telescope or microscope.
Philosophy of science, laments Pagels, has fallen on hard times. Once scientists like Einstein and Max Born hotly debated issues in the philosophy of science; today scientists have withdrawn from theoretical philosophical issues and have instead expressed their philosophical interests in the form of social activism. Part of the reason for this is the turn philosophy of science took down the blind alley of conceptual relativism in the 1960s under the influence of Thomas Kuhns well-known view of the history of science as a series of revolutionary paradigm shifts. Remarkably, a lot of what was said in Kuhns books, observes Pagels, I could easily imagine describing the high fashion world as well as science. There are periods of normal fashion changes, but sometimes there are major paradigm shifts brought about by a younger generation. The reason why this content substitution works so well is that Kuhns extreme conceptual relativism leaves him blind to the fact that the primary selective pressure on scientific ideas is something external, to science the invariant order of nature. It is also why philosophy of science is today in danger of drifting off into a kind of narrative literature, which Pagels dismisses as a high-level but empty intellectual vision.
Chaos theory is still too new to fully fathom its implications for our philosophic and scientific views of the universe. Questioned by Napoleon about the absence of God from his huge work on the system of the world, Laplace, known for his aggressive atheism, boasted, Sire, I have no need of the hypothesis. Poincares discovery of chaos, however, uncovered a flaw at the center of Laplaces dream of a god-less, deterministic, predictable universe. Nothing short of an intellect who could know the initial conditions of the universe with infinite precision could serve as Laplaces Demiurge. And far from dispelling the notion of order, scientists discovered that deep within chaos there lurks a new type of order, infinitely more intricate than the forms of classical Euclidian geometry. In a famous letter to Born, Einstein complained: You believe in a God who plays dice, and I in complete law and order. The discovery of deterministic chaos, however, calls into question the very distinction Einstein was trying to invoke. It turns out, paradoxically, that God can play dice in a universe of complete law and order. Perhaps we should recall Wolfgang Paulis famous retort to Einstein: Dont tell God what to do with the universe.